Numerical Methods for High-Speed Flows

We review numerical methods for direct numerical simulation (DNS) and large-eddy simulation (LES) of turbulent compressible flow in the presence of shock waves. Ideal numerical methods should be accurate and free from numerical dissipation in smooth parts of the flow, and at the same time they must robustly capture shock waves without significant Gibbs ringing, which may lead to nonlinear instability. Adapting to these conflicting goals leads to the design of strongly nonlinear numerical schemes that depend on the geometrical properties of the solution. For low-dissipation methods for smooth flows, numerical stability can be based on physical conservation principles for kinetic energy and/or entropy. Shock-capturing requires the addition of artificial dissipation, in more or less explicit form, as a surrogate for physical viscosity, to obtain nonoscillatory transitions. Methods suitable for both smooth and shocked flows are discussed, and the potential for hybridization is highlighted. Examples of the application of advanced algorithms to DNS/LES of turbulent, compressible flows are presented.

[1]  Andrew W. Cook,et al.  Short Note: Hyperviscosity for shock-turbulence interactions , 2005 .

[2]  Jung Yul Yoo,et al.  Discretization errors in large eddy simulation: on the suitability of centered and upwind-biased compact difference schemes , 2004 .

[3]  R. Courant,et al.  On the solution of nonlinear hyperbolic differential equations by finite differences , 1952 .

[4]  Johan Larsson,et al.  Stability criteria for hybrid difference methods , 2008, J. Comput. Phys..

[5]  Holger Babinsky,et al.  Shock Wave-Boundary-Layer Interactions , 2014 .

[6]  D. Pullin,et al.  Hybrid tuned center-difference-WENO method for large eddy simulations in the presence of strong shocks , 2004 .

[7]  A. Harten,et al.  The artificial compression method for computation of shocks and contact discontinuities: III. Self , 1978 .

[8]  A. Sadiki,et al.  A digital filter based generation of inflow data for spatially developing direct numerical or large eddy simulations , 2003 .

[9]  Vincent Guinot,et al.  High-Order Fluxes for Conservative Skew-Symmetric-like Schemes in Structured Meshes , 2000 .

[10]  Parviz Moin,et al.  Interaction of isotropic turbulence with shock waves: effect of shock strength , 1997, Journal of Fluid Mechanics.

[11]  Chi-Wang Shu.  Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

[12]  W. Cabot,et al.  A high-wavenumber viscosity for high-resolution numerical methods , 2004 .

[13]  Minwei Wu,et al.  Direct numerical simulation of supersonic turbulent boundary layer over a compression ramp , 2007 .

[14]  Sergio Pirozzoli,et al.  Direct numerical simulation of impinging shock wave/turbulent boundary layer interaction at M=2.25 , 2006 .

[15]  Gianluca Iaccarino,et al.  IMMERSED BOUNDARY METHODS , 2005 .

[16]  H. C. Yee,et al.  Entropy Splitting for High Order Numerical Simulation of Compressible Turbulence , 2002 .

[17]  E. Tadmor.  Skew-selfadjoint form for systems of conservation laws , 1984 .

[18]  Xiaogang Deng,et al.  Developing high-order weighted compact nonlinear schemes , 2000 .

[19]  Joel H. Ferziger,et al.  A robust high-order compact method for large eddy simulation , 2003 .

[20]  Matteo Bernardini,et al.  A general strategy for the optimization of Runge-Kutta schemes for wave propagation phenomena , 2009, J. Comput. Phys..

[21]  R. D. Richtmyer,et al.  Difference methods for initial-value problems , 1959 .

[22]  Pierre Sagaut,et al.  A Class of Explicit ENO Filters with Application to Unsteady Flows , 2001 .

[23]  Michael Dumbser,et al.  Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems , 2007, J. Comput. Phys..

[24]  T. Lund,et al.  Generation of Turbulent Inflow Data for Spatially-Developing Boundary Layer Simulations , 1998 .

[25]  Kenneth J. Ruschak,et al.  MODELING ARTIFICIAL BOUNDARY CONDITIONS FOR COMPRESSIBLE FLOW , 2005 .

[26]  Christophe Bailly,et al.  A shock-capturing methodology based on adaptative spatial filtering for high-order non-linear computations , 2009, J. Comput. Phys..

[27]  Sergio Pirozzoli,et al.  Generalized conservative approximations of split convective derivative operators , 2010, J. Comput. Phys..

[28]  N. Sandham,et al.  Large-eddy simulation of transonic turbulent flow over a bump , 2003 .

[29]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[30]  Gunilla Kreiss,et al.  A Remark on Numerical Errors Downstream of Slightly Viscous Shocks , 1999 .

[31]  Xu-Dong Liu,et al.  Solution of Two-Dimensional Riemann Problems of Gas Dynamics by Positive Schemes , 1998, SIAM J. Sci. Comput..

[32]  A. Harten,et al.  The artificial compression method for computation of shocks and contact discontinuities. I - Single conservation laws , 1977 .

[33]  Parviz Moin,et al.  EDDY SHOCKLETS IN DECAYING COMPRESSIBLE TURBULENCE , 1991 .

[34]  Pramod K. Subbareddy,et al.  A fully discrete, kinetic energy consistent finite-volume scheme for compressible flows , 2009, J. Comput. Phys..

[35]  G. A. Gerolymos,et al.  Very-high-order weno schemes , 2009, J. Comput. Phys..

[36]  Parviz Moin,et al.  Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves , 2010, J. Comput. Phys..

[37]  J. Steger,et al.  Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods , 1981 .

[38]  Mark H. Carpenter,et al.  Computational Considerations for the Simulation of Shock-Induced Sound , 1998, SIAM J. Sci. Comput..

[39]  Culbert B. Laney,et al.  Computational Gasdynamics: Waves , 1998 .

[40]  Yohei Morinishi,et al.  Skew-symmetric form of convective terms and fully conservative finite difference schemes for variable density low-Mach number flows , 2010, J. Comput. Phys..

[41]  John A. Ekaterinaris,et al.  High-order accurate, low numerical diffusion methods for aerodynamics , 2005 .

[42]  Miguel R. Visbal,et al.  High-Order-Accurate Methods for Complex Unsteady Subsonic Flows , 1999 .

[43]  W. Habashi,et al.  2N-Storage Low Dissipation and Dispersion Runge-Kutta Schemes for Computational Acoustics , 1998 .

[44]  H. C. Yee,et al.  A class of high resolution explicit and implicit shock-capturing methods , 1989 .

[45]  A. Harten.  On the symmetric form of systems of conservation laws with entropy , 1983 .

[46]  P. Sagaut,et al.  On the Use of Shock-Capturing Schemes for Large-Eddy Simulation , 1999 .

[47]  A. Sidorenko,et al.  Investigation by Particle Image Velocimetry Measurements of Oblique Shock Reflection with Separation , 2008 .

[48]  Eitan Tadmor,et al.  Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems , 2003, Acta Numerica.

[49]  Ravi Samtaney,et al.  Direct numerical simulation of decaying compressible turbulence and shocklet statistics , 2001 .

[50]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

[51]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

[52]  Sanjiva K. Lele,et al.  A modified artificial viscosity approach for compressible turbulence simulations , 2008, J. Comput. Phys..

[53]  Gino Moretti,et al.  COMPUTATION OF FLOWS WITH SHOCKS , 1987 .

[54]  A. Jameson,et al.  Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes , 1981 .

[55]  Shih-Hsien Yu,et al.  Discrete Shocks for Finite Difference Approximations to Scalar Conservation Laws , 1998 .

[56]  Eleuterio F. Toro,et al.  Finite-volume WENO schemes for three-dimensional conservation laws , 2004 .

[57]  Andrew W. Cook,et al.  Artificial Fluid Properties for Large-Eddy Simulation of Compressible Turbulent Mixing , 2007 .

[58]  J. Bowles,et al.  Fourier Analysis of Numerical Approximations of Hyperbolic Equations , 1987 .

[59]  Marcel Vinokur,et al.  Conservation equations of gasdynamics in curvilinear coordinate systems , 1974 .

[60]  Miguel R. Visbal,et al.  Shock Capturing Using Compact-Differencing-Based Methods , 2005 .

[61]  Sergio Pirozzoli,et al.  Conservative Hybrid Compact-WENO Schemes for Shock-Turbulence Interaction , 2002 .

[62]  Neil D. Sandham,et al.  Large-eddy simulation of low-frequency unsteadiness in a turbulent shock-induced separation bubble , 2009 .

[63]  Eric Johnsen,et al.  Implementation of WENO schemes in compressible multicomponent flow problems , 2005, J. Comput. Phys..

[64]  Eitan Tadmor,et al.  The numerical viscosity of entropy stable schemes for systems of conservation laws. I , 1987 .

[65]  Parviz Moin,et al.  Higher entropy conservation and numerical stability of compressible turbulence simulations , 2004 .

[66]  E. Tadmor,et al.  Convergence of spectral methods for nonlinear conservation laws. Final report , 1989 .

[67]  Neil D. Sandham,et al.  Low-Dissipative High-Order Shock-Capturing Methods Using Characteristic-Based Filters , 1999 .

[68]  B. Strand.  Summation by parts for finite difference approximations for d/dx , 1994 .

[69]  Eleuterio F. Toro,et al.  Towards Very High Order Godunov Schemes , 2001 .

[70]  V. Gregory Weirs,et al.  Optimization of weighted ENO schemes for DNS of compressible turbulence , 1997 .

[71]  Russell M. Cummings,et al.  Critical Hypersonic Aerothermodynamic Phenomena , 2006 .

[72]  C. Bogey,et al.  Low-dissipation and low-dispersion fourth-order Runge–Kutta algorithm , 2006 .

[73]  Pierre Sagaut,et al.  Turbulent Inflow Conditions for Large-Eddy-Simulation of Compressible Wall-Bounded Flows , 2004 .

[74]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[75]  Sanjiva K. Lele,et al.  Compressibility Effects on Turbulence , 1994 .

[76]  Nikolaus A. Adams,et al.  A conservative interface method for compressible flows , 2006, J. Comput. Phys..

[77]  Soshi Kawai,et al.  Localized artificial diffusivity scheme for discontinuity capturing on curvilinear meshes , 2008, J. Comput. Phys..

[78]  Paolo Orlandi,et al.  Fluid Flow Phenomena , 2000 .

[79]  Antony Jameson,et al.  The Construction of Discretely Conservative Finite Volume Schemes that Also Globally Conserve Energy or Entropy , 2008, J. Sci. Comput..

[80]  Sanjiva K. Lele,et al.  An artificial nonlinear diffusivity method for supersonic reacting flows with shocks , 2005, J. Comput. Phys..

[81]  P. Olsson,et al.  A High Order Finite Difference Scheme with Sharp Shock Resolution for the Euler Equations , 1996 .

[82]  Nikolaus A. Adams,et al.  Direct simulation of the turbulent boundary layer along a compression ramp at M = 3 and Reθ = 1685 , 2000, Journal of Fluid Mechanics.

[83]  P. Moin,et al.  Direct numerical simulation of transition and turbulence in a spatially evolving boundary layer , 1991 .

[84]  P. Moin,et al.  DIRECT NUMERICAL SIMULATION: A Tool in Turbulence Research , 1998 .

[85]  S. Lele,et al.  A Numerical Study of Shock-Associated Noise , 2002 .

[86]  Eckart Meiburg,et al.  A numerical study of the convergence properties of ENO schemes , 1990 .

[87]  T. Colonius,et al.  Computational aeroacoustics: progress on nonlinear problems of sound generation , 2004 .

[88]  Z. Wang.  High-order methods for the Euler and Navier–Stokes equations on unstructured grids , 2007 .

[89]  C. Bogey,et al.  A family of low dispersive and low dissipative explicit schemes for flow and noise computations , 2004 .

[90]  M. Pino Martin,et al.  Assessment of inflow boundary conditions for compressible turbulent boundary layers , 2004 .

[91]  Nikolaus A. Adams,et al.  A High-Resolution Hybrid Compact-ENO Scheme for Shock-Turbulence Interaction Problems , 1996 .

[92]  Olsson,et al.  SUMMATION BY PARTS, PROJECTIONS, AND STABILITY. I , 2010 .

[93]  Björn Sjögreen,et al.  Development of low dissipative high order filter schemes for multiscale Navier-Stokes/MHD systems , 2006, J. Comput. Phys..

[94]  S. Osher,et al.  Weighted essentially non-oscillatory schemes , 1994 .

[95]  H. Huynh,et al.  Accurate Monotonicity-Preserving Schemes with Runge-Kutta Time Stepping , 1997 .

[96]  Christopher A. Kennedy,et al.  Reduced aliasing formulations of the convective terms within the Navier-Stokes equations for a compressible fluid , 2008, J. Comput. Phys..

[97]  Sanjiva K. Lele,et al.  Direct numerical simulations of canonical shock/turbulence interaction , 2008, Proceeding of Sixth International Symposium on Turbulence and Shear Flow Phenomena.

[98]  R. D. Richtmyer,et al.  A Method for the Numerical Calculation of Hydrodynamic Shocks , 1950 .

[99]  M.Y. Hussaini,et al.  Low-Dissipation and Low-Dispersion Runge-Kutta Schemes for Computational Acoustics , 1994 .

[100]  Nikolaus A. Adams,et al.  A Subgrid-Scale Deconvolution Approach for Shock Capturing , 2002 .

[101]  J. C. Kok,et al.  A high-order low-dispersion symmetry-preserving finite-volume method for compressible flow on curvilinear grids , 2009, J. Comput. Phys..

[102]  Chi-Wang Shu,et al.  Monotonicity Preserving Weighted Essentially Non-oscillatory Schemes with Increasingly High Order of Accuracy , 2000 .

[103]  Michael Dumbser,et al.  A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes , 2008, J. Comput. Phys..

[104]  Matteo Bernardini,et al.  Direct numerical simulation of transonic shock/boundary layer interaction under conditions of incipient separation , 2010, Journal of Fluid Mechanics.

[105]  Miguel R. Visbal,et al.  Large-Eddy Simulation of Supersonic Compression-Ramp Flow by High-Order Method , 2001 .

[106]  Chi-Wang Shu.  Numerical experiments on the accuracy of ENO and modified ENO schemes , 1990 .

[107]  Joel H. Ferziger,et al.  Numerical simulation of a compressible, homogeneous, turbulent shear flow , 1981 .

[108]  Margot Gerritsen,et al.  Designing an efficient solution strategy for fluid flows. 1. A stable high order finite difference scheme and sharp shock resolution for the Euler equations , 1996 .

[109]  Joel H. Ferziger,et al.  Improved Methods for Large Eddy Simulations of Turbulence , 1979 .

[110]  J. Blair Perot,et al.  Discrete Conservation Properties of Unstructured Mesh Schemes , 2011 .

[111]  Parviz Moin,et al.  Suitability of artificial bulk viscosity for large-eddy simulation of turbulent flows with shocks , 2009, J. Comput. Phys..

[112]  Soshi Kawai,et al.  Large-Eddy Simulation of Jet Mixing in Supersonic Crossflows , 2010 .

[113]  P. Sagaut,et al.  Large-eddy simulation of the shock/boundary layer interaction , 2001 .

[114]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[115]  M. Pino Martín,et al.  Optimization of nonlinear error for weighted essentially non-oscillatory methods in direct numerical simulations of compressible turbulence , 2007, J. Comput. Phys..

[116]  A. Majda.  Compressible fluid flow and systems of conservation laws in several space variables , 1984 .

[117]  P. Lax.  Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves , 1987 .

[118]  Ben-yu Guo,et al.  Spectral Vanishing Viscosity Method For Nonlinear Conservation Laws , 2001, SIAM J. Numer. Anal..

[119]  Antony Jameson,et al.  Formulation of Kinetic Energy Preserving Conservative Schemes for Gas Dynamics and Direct Numerical Simulation of One-Dimensional Viscous Compressible Flow in a Shock Tube Using Entropy and Kinetic Energy Preserving Schemes , 2008, J. Sci. Comput..

[120]  Chaowei Hu,et al.  No . 98-32 Weighted Essentially Non-Oscillatory Schemes on Triangular Meshes , 1998 .

[121]  Gino Moretti,et al.  The λ-scheme , 1979 .

[122]  C. Hirsch,et al.  Numerical Computation of Internal and External Flows. By C. HIRSCH. Wiley. Vol. 1, Fundamentals of Numerical Discretization. 1988. 515 pp. £60. Vol. 2, Computational Methods for Inviscid and Viscous Flows. 1990, 691 pp. £65. , 1991, Journal of Fluid Mechanics.

[123]  Douglas K. Lilly,et al.  ON THE COMPUTATIONAL STABILITY OF NUMERICAL SOLUTIONS OF TIME-DEPENDENT NON-LINEAR GEOPHYSICAL FLUID DYNAMICS PROBLEMS , 1965 .

[124]  Paolo Orlandi,et al.  Fluid Flow Phenomena: A Numerical Toolkit , 1999 .

[125]  Rajat Mittal,et al.  A sharp interface immersed boundary method for compressible viscous flows , 2007, J. Comput. Phys..

[126]  P. Moin,et al.  Effect of numerical dissipation on the predicted spectra for compressible turbulence , 2022 .

[127]  Dale Pullin,et al.  Large-eddy simulation and multiscale modelling of a Richtmyer–Meshkov instability with reshock , 2006, Journal of Fluid Mechanics.

[128]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[129]  With Invariant Submanifolds,et al.  Systems of Conservation Laws , 2009 .

[130]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[131]  S. Balachandar,et al.  Mechanisms for generating coherent packets of hairpin vortices in channel flow , 1999, Journal of Fluid Mechanics.

[132]  V. Gregory Weirs,et al.  A bandwidth-optimized WENO scheme for the effective direct numerical simulation of compressible turbulence , 2006, J. Comput. Phys..

[133]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[134]  P. Olsson.  Summation by parts, projections, and stability. II , 1995 .

[135]  Miguel R. Visbal,et al.  On the use of higher-order finite-difference schemes on curvilinear and deforming meshes , 2002 .

[136]  Sergio Pirozzoli,et al.  On the spectral properties of shock-capturing schemes , 2006, J. Comput. Phys..

[137]  C. Tam,et al.  Dispersion-relation-preserving finite difference schemes for computational acoustics , 1993 .

[138]  Björn Sjögreen,et al.  Multiresolution Wavelet Based Adaptive Numerical Dissipation Control for Shock-Turbulence Computations , 2001 .

[139]  A. Harten.  ENO schemes with subcell resolution , 1989 .

[140]  Sangsan Lee,et al.  Large eddy simulation of shock turbulence interaction , 1993 .

[141]  Pierre Sagaut,et al.  Generation of synthetic turbulent inflow data for large eddy simulation of spatially evolving wall-bounded flows , 2009 .

[142]  Eleuterio F. Toro,et al.  ADER schemes for three-dimensional non-linear hyperbolic systems , 2005 .

[143]  Chi-Wang Shu,et al.  High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems , 2009, SIAM Rev..

[144]  Yuxin Ren,et al.  A characteristic-wise hybrid compact-WENO scheme for solving hyperbolic conservation laws , 2003 .

[145]  S. Osher,et al.  Uniformly high order accuracy essentially non-oscillatory schemes III , 1987 .

[146]  R. Lewis,et al.  Low-storage, Explicit Runge-Kutta Schemes for the Compressible Navier-Stokes Equations , 2000 .

[147]  H. C. Yee,et al.  Multiresolution Wavelet Based Adaptive Numerical Dissipation Control for Shock-Turbulence Computations , 2001 .

[148]  S. Lele.  Compact finite difference schemes with spectral-like resolution , 1992 .

[149]  Pietro De Palma,et al.  An immersed boundary method for compressible flows using local grid refinement , 2007, J. Comput. Phys..

[150]  Philip L. Roe,et al.  On Postshock Oscillations Due to Shock Capturing Schemes in Unsteady Flows , 1997 .

[151]  David W. Zingg,et al.  An optimized finite-difference scheme for wave propagation problems , 1993 .

[152]  Holger Foysi,et al.  Compressibility effects and turbulence scalings in supersonic channel flow , 2004, Journal of Fluid Mechanics.

[153]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[154]  Joseph Oliger,et al.  Energy and Maximum Norm Es-timates for Nonlinear Conservation Laws , 1994 .

[155]  Gregory A. Blaisdell,et al.  The effect of the formulation of nonlinear terms on aliasing errors in spectral methods , 1996 .

[156]  Matteo Bernardini,et al.  Characterization of coherent vortical structures in a supersonic turbulent boundary layer , 2008, Journal of Fluid Mechanics.

[157]  Björn Sjögreen,et al.  The Convergence Rate of Finite Difference Schemes in the Presence of Shocks , 1998 .